Question

The law of motion in a straight line $x=\frac{1}{2}vt$, then acceleration is

• A. Constant
• B. Proportional to $t$
• C. Proportional to $v$
• D. Proportional to $s$

Answer 1

The correct option is A

Constant

Step 1: Given data:

The law of motion in a straight line $x=\frac{1}{2}vt$.

where $v=$velocity and $t=$time.

Step 2: Formula used:

Know that the law of motion in a straight line,

$x=\frac{1}{2}vt$……(1)

Velocity is the rate of change of displacement that is,

$v=\frac{dx}{dt}$….(2)

Acceleration is defined as-

$a=\frac{d^{2}x}{d{t}^2}$

Now put the value of velocity from equation (2) to equation (1)-

$x=\frac{1}{2}\frac{dx}{dt}t$

On arranging,

$\frac{2dt}{t}=\frac{dx}{x}$…..(3)

Step 3: Integration of the function:

Now integrate the function (3)-

$$\begin{gathered} \Rightarrow 2\int \frac{dt}{t}=\int \frac{dx}{x} \\ \Rightarrow 2\log \left(t\right)+\log \left(c\right)=\log \left(x\right) \\ \Rightarrow \log \left(t^2·c\right)=\log \left(x\right) \\ \Rightarrow x=\left(t^2·c\right)............................\left(3\right) \end{gathered}$$

Now differentiate the function (3) to obtain acceleration-

$$\begin{gathered} \frac{dx}{dt}=2tc \\ \frac{d^{2}x}{d{t}^2}=2c \\ a=cons\tan t \end{gathered}$$

Therefore the acceleration is constant.

Hence, option A is the correct answer.